Complete Congruences of Completely Distributive Lattices

All the binomial lattices embed into \(Q_{\vee }(\mathbb {I})\) , the complete lattice of sup-preserving endomaps of the unit interval—whose elements can be seen as continuous monotone paths from (0, 0) to (1, 1). This lattice is completely distributive. We give a general description of the complete congruences of completely distributive lattice s by means of an interior operator on the collection of closed subsets of an associated topological space. In particular, we show that these form a frame. We give a description of this frame for the unit interval lattice, showing that it is not a Boolean algebra nor a (co)spatial frame. For \(Q_{\vee }(\mathbb {I})\) , we give a geometrical interpretation of these congruences by means of directed homotopies.